Medial category
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In mathematics, the medial category Med, that is, the category of medial magmas has as objects sets with a medial binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense).
The category Med has direct products, so the concept of a medial magma object (internal binary operation) makes sense. As a result, Med has all its objects as medial objects, and this characterizes it.
There is an inclusion functor from Set to Med as trivial magmas, with operations being the right projections
- (x, y) → y.
An injective endomorphism can be extended to an automorphism of a magma extension — the colimit of the constant sequence of the endomorphism.